On the Lang-Trotter conjecture for a class of non-generic abelian surfaces

In the present article, we formulate a conjectural uniform error term in the Chebotarev-Sato-Tate distribution for abelian surfaces Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}-isogenous to a product of not Q<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb {Q}}$$\end{document}-isogenous non-CM-elliptic curves, established by the author in Amri (Eur J Math, 2023. https://doi.org/10.1007/s40879-023-00682-5, Theorem 1.1). As a consequence, we provide a conditional direct proof to the generalized Lang-Trotter conjecture recently formulated and studied in Chen et al. (Ramanujan J, 2022).